How to compute upper limits?

What you will learn

You will learn how to determine the upper flux limit for a source using the ctulimit tool.

You determine upper limits for gamma-ray sources using the ctulimit tool.

Suppose you want to know the upper limit on the gamma-ray flux for a putative source at position \(l=0.1^\circ\) and \(b=0.4^\circ\). For this you have to add a test source with the respective coordinates to the model definition file. To do this, copy the 1DC model definition file

$ cp $CTOOLS/share/models/1dc_howto.xml ulimit_model.xml

to ulimit_model.xml and add the following source to it:

  <source name="Test" type="PointSource">
    <spectrum type="PowerLaw">
      <parameter name="Prefactor"   value="1" scale="1.0e-20" min="0"          free="1"/>
      <parameter name="Index"       value="1" scale="-2.48"   min="-5" max="5" free="0"/>
      <parameter name="PivotEnergy" value="1" scale="300000"                   free="0"/>
    </spectrum>
    <spatialModel type="PointSource">
      <parameter name="GLON" value="0.1" scale="1" free="0"/>
      <parameter name="GLAT" value="0.4" scale="1" free="0"/>
    </spatialModel>
  </source>

Now run the ctulimit tool as follows:

$ ctulimit
Input event list, counts cube or observation definition XML file [events.fits] cntcube.fits
Input exposure cube file [NONE] expcube.fits
Input PSF cube file [NONE] psfcube.fits
Input background cube file [NONE] bkgcube.fits
Source of interest [Crab] Test
Input model definition XML file [$CTOOLS/share/models/crab.xml] ulimit_model.xml

After the run, the upper gamma-ray flux limit for this test source can be extracted from the ctulimit log file:

2019-04-04T10:31:20: +=====================+
2019-04-04T10:31:20: | Upper limit results |
2019-04-04T10:31:20: +=====================+
2019-04-04T10:31:20:  Differential flux limit ...: 1.3961626479978e-20 ph/cm2/s/MeV at 1 TeV
2019-04-04T10:31:20:  Integral flux limit .......: 9.42318774389192e-15 ph/cm2/s within [1-100] TeV
2019-04-04T10:31:20:  Energy flux limit .........: 4.14922476753972e-14 erg/cm2/s within [1-100] TeV

Warning

It is important to fix the spectral index of the test source in the upper limit computation, since for a source that is not detected, the index of the spectral law cannot be constrained. In other words, you have to make an assumption about the spectral energy density distribution when computing the upper flux limit of a source. The spectral index is fixed by setting free="0" for the Index parameter.